In the printing industry, color halftoning and color management are subjects that need to be continuously improved in order to get newer and more accurate ways to predict and reproduce color in a variety of media. In particular, color inkjet printers are devices where research and development efforts have been concentrated in the past several years. Inkjet printing involves a variety of physical phenomena, such as light scattering and ink spreading, which make accurate color prediction a difficult task.
The simplest approach to characterizing a color printer is to empirically take several calorimetric measurements over a wide range of color patches in a grid-like fashion. Interpolation of such data can be done to estimate the rest of the model, where the amount of error depends on the number of grid points taken into account. For example, in a Cyan, Magenta and Yellow (CMY) printer, small errors could be attained with around 8000 measurements. However, the amount of memory that is required to store this type of model is a major drawback. In addition, since the model characterizes a particular printer and media, this method lacks flexibility because a new set of colorimetric data is needed for each printer. An alternative approach that considerably reduces the number of parameters to be stored is to characterize the printing process with a spectral model.
Perhaps the earliest approach to a spectral model for monochrome halftoning was proposed by Murray and Davies in 1936 as a linear combination of reflectance and area of coverage. The Murray-Davies model was further extended by Neugebauer in 1937 to color halftones. The Neugebauer equations are built on the work of Demichel and consider a given area of photomechanical printing to be formed by eight different areas, each one covered by a different color: yellow, cyan, magenta, blue, green, red, black and white. The Demichel equations represent the corresponding eight fractional areas, which are fused by the eye in order to produce the sensation of a single color. The Neugebauer equations express the red, green and blue (RGB) tristimulus response in terms of such fractional areas and their reflectances. Denoting ai and ri(λ) as the fractional area of the ith colorant (i.e., Demichel coefficients) and the reflectance, respectively, the spectral Neugebauer equations for the total reflectance r(λ) are given by
                                          r            ⁡                          (              λ              )                                =                                    ∑                              i                =                1                            8                        ⁢                                          a                i                            ⁢                                                r                  i                                ⁡                                  (                  λ                  )                                                                    ,                            (        1        )            constrained to
            ∑              i        =        1            8        ⁢          a      i        =  1.That is, the total reflectance r(λ) is an additive mixture of the reflectances ri(λ) of the eight fractional areas ai.
Naturally, the reflectance is a continuous function of the wavelength. However, in practice, sampled values between 400 nm and 700 nm are used in most models since even modem spectrophotometers are just able to measure the reflection characteristics of an object at discrete wavelengths. The sample reflectance matrix Rc of a given color is then a set of reflectances r=[r(λ0), r(λ1), r(λ2), . . . , r(λk)] at discrete wavelengths λi, as shown in FIG. 1. Rewriting Eq. (1) in matrix form givesr=aRcT,  (2)where ri=[ri(λ0), . . . , ri(λk)], a=[a1, . . . , a8], and Rc=[r1T |r2T|, . . . , |r8T].
More recent approaches to spectral modeling for printer characterization make an empirical correction to Neugebauer's theory by finding a linear relationship between the densities of RGB and CMY. An example of this approach was proposed by Clapper in 1961. By using a second order transformation between the densities of RGB in the three color combinations and the principal densities of the individual primary color inks (i.e., CMY), the inaccuracies due to non-linear effects were reduced. Higher order equations may be used to reduce the error even further, as was shown by Heuberger et al in 1992.
However, a completely different approach from the above-discussed background art that reduces both the number of measurements and the number of parameters required is needed to more efficiently characterize the spectral model.